This thesis is concerned with so-called higher derivative terms which arise in low energy approximations to certain physical models. In particular, the aim is to investigate the role that such terms play in low energy N=2 supersymmetric gauge theories in 4 dimensions, with gauge group SU(2).Chapter one serves as an introduction to the notions of supersymmetry and superfields. The problem of constructing an effective action which describes the low energy dynamics is introduced, and the construction of the Wilsonian action in terms of light and heavy modes is developed. The concept on a derivative expansion is also described. Chapter two introduces N=2 supersymmetric gauge theories with spontaneous symmetry breaking. It is observed that such systems always have a Bogomolnyi bound, and the consequences are discussed. We then develop a derivative expansion of this system in terms of N=2 superfields, drawing particular attention to the next-to- leading order derivative term (that is, those with 4 derivatives/8 fermions). The duality properties of such a term are reviewed, and their impact on the mass formula discussed. Conclusions are drawn as to their influence on the results of Seiberg and Witten. Chapter three deals with a non-renormalisation theorem for the next-to-leading order higher derivative term proposed by Dine and Seiberg. This states that instanton contributions to such a term in massless N=2 SU(N(_c)) gauge theories vanish when the number of flavours N(_f) = 2N(_c). We prove this result using the ADHM formalism for multi-instantons in the case N(_c) = 2.